ÖPVERSIGT AF K. VETENSK.-AKAD. FÖRHANDLINGAR 1899, N:0 7. 679 



Before proceeding further in the development of the ex- 

 pression for the constant K^-, we will give proper forms to the 

 constants of which jSr„j is a function, viz. Cm and S^, and for 

 which reason we will prove certain necessary relations. At first 



we will take the sum ^ A cos rr/., where we suppose that A has 



ra = l 



two different values, — one for an odd factor n to z, which we 

 will design with A^ and the other for an even n, viz. A>,. Thus 

 if m is odd we get 



rii 



^ A COS iv/.—A^[co& mx + cos {m — 2)/t + ... + cos 3z + cos x] + 



n = l 



+ Å^Q.o?>{in — l)>t+cos(7?2., — 3)/. + . . + cos4>t + cos2x] 



. TO + 1 7rt 4- 1 . ?7l — 1 m + 1 



sin —g- /t cos ^ö— >'- sm — ^— /. cos — ^— x 



= ^1 — ^— ^ ^ + A, -—. ~ (22) 



'■ sm ■/. " sm X 



w = l 



cos nx 



»71 + 1 »I + 1 



sm — f;— X cos — 7^— X 



sm X 



^ . ??i + 1 . ??i + 1 



™ sm^-xsm^-x 



^ A sm ?r/t = ^j — 



+ ^2 



sin 



m 



2 



1 

 - X cos 



7)1 + 1 



2 "''^ 



(23) 









sin X 







sin 



m 





1 . 

 - X sin 



m + 1 



2 "- 





4-^2 











(24) 













M=l 



Sin X sm X 



2(— i)"^si 



sm mx = 



. 7W + 1 . ?)J i' 1 .7» 1 . 7rt + 1 



sm — ^ — X sm -^ — X sm — ^ — x sin — ^r^ x 



= — ^, — -. + A. "—. ^ (25) 



sm X " sin X 



If m even 



,„ . in m ■ m to + 2 



™ Sin -^ X cos ^ X sin -;;5- x cos -^- x 



>, ^ cos nx = + J, —. ~ + J. "—. =-— (26) 



n = \ 



sm X " sm X 



„, .TOTO . TO TO + 2 



™ sin-^xcos-^x sin^xcos— ^— X 



2(-l)Mcoswx = — J. —. —+A, ^—. ~ (27) 



~ ' sm X ^ sm X ^ 



